The human global population has expanded more than 1000-fold in the last 400 generations, resulting in a state that is profoundly out of equilibrium with respect to genetic variation. The recent growth produces a large excess of rare variation, which has important consequences for finding genes that underlie complex disease risk. Our overall objective is to develop and test methods of population genetic analysis to understand the role of rapid population expansion in shaping patterns of genetic variation. In Aim 1 we will develop theoretical approaches to understand how and why explosive growth impacts patterns of genetic variation. We will also derive the analytical implications of using samples that are so large as to violate assumptions of the neutral coalescent. We have shown how large samples can result in multiple mergers, and so both rapid growth and large sample sizes distort the topology of the gene genealogies of a sample so as to make standard coalescent theory invalid. We will replace this with new methods that generate the appropriate sample site frequency spectrum under models with both rapid growth and large samples. Given large data sets, we want to make inference about population genetic parameters, and such estimates generally require an appropriate model relating population size and mutation rates to levels of variation. In Aim 2 we will develop novel statistical and computational inference methods to accommodate growing populations and apply them to large-scale data. We will thoroughly test our inference methods using simulation data generated under appropriate demographic models. This aim will generate novel software packages with broad utility for the community. In Aim 3 we will learn how natural selection in a rapidly growing population impacts population genetic variation and the architecture of complex traits. This goal will be accomplished through extensive forward-in-time simulations. Among other things, results will tell us conditions under which rapid growth inflates the individual mutation load. By developing an understanding of the way that such rapid growth has impacted genetic variation in humans, we anticipate that these results will provide a more accurate picture of the expected genetic architecture of disease risk, which will in turn guide methods for improved association testing.